KMS States of Self-Similar $k$-Graph C*-Algebras

TL;DR

This paper characterizes the structure of KMS states for certain self-similar $k$-graph C*-algebras, linking it to the periodicity group and Perron-Frobenius eigenvector properties.

Contribution

It provides a detailed description of the KMS simplex for these algebras under specific self-similar action conditions, extending understanding of their state space structure.

Findings

KMS simplex is either empty or isomorphic to the tracial state space of the periodicity group C*-algebra.

The structure depends on whether the Perron-Frobenius eigenvector preserves the group action.

Several important classes of examples are explicitly constructed and analyzed.

Abstract

Let $G$ be a countable discrete amenable group, and $\Lambda$ be a strongly connected finite $k$-graph. If $(G,\Lambda)$ is a pseudo free and locally faithful self-similar action which satisfies the finite-state condition, then the structure of the KMS simplex of the C*-algebra $\O_{G,\Lambda}$ associated to $(G,\Lambda)$ is described: it is either empty or affinely isomorphic to the tracial state space of the C*-algebra of the periodicity group $\Per_{G,\Lambda}$ of $(G,\Lambda)$, depending on whether the Perron-Frobenius eigenvector of $\Lambda$ preserves the $G$-action. As applications of our main results, we also exhibit several classes of important examples.